ANNAPOLIS, Sept. 19-An Anne Arundel County fire engine was en route to a fire recently when it rounded a corner and its doors popped open because the engine's body was in such bad shape it could not handle the stress, fire officials say. No one was injured, but county fire officials used that example in their efforts to persuade the county government that they need new fire trucks.

Today, the county council listened to the firefighters, and approved a $1.76 million contract to buy eight new fire engines and one new ladder truck in a lease-purchase agreement with First National Bank of Maryland. The council's approval was significant considering the fiscal problems the county has faced. This year, the council was forced to raise the county property tax rate 37 cents to $2.68 per $100 of assessed value.

Under the agreement, the county will make 10 payments of $176,000 each over the next five years to the bank, paying 8.5 percent interest. The bank will retain ownership of the trucks until the final payment is made although the trucks will be delivered in 90 days.

Robert Dvorak, county fire administrator, told the council that all the new trucks were needed now because the county is several years behind in replacing old, worn-out trucks. The county should be replacing four to five trucks a year, Dvorak said, but it has not replaced any since 1981. Most of the fleet's five ladder trucks and 64 engines were bought in the early and middle 1960s.

The council's seven members, some of whom criticized borrowing money to buy the trucks, all voted for the measure.

Anne Arundel County had not budgeted for replacement of county equipment in the last two fiscal years that ended June 30. County Executive O. James Lighthizer said his predecessor, Robert A. Pascal, attempted to keep the tax rate "artificially low" preceding his bid for governor last year.

In other action today, the council approved a final version of a noise bill that outlaws "hooting, singing or whistling" between the hours of 11 p.m. and 7 a.m. The original bill, introduced by Councilman Edward C. Ahern (D-Pasadena), was vetoed by Lighthizer because, the executive said, it was unconstitutional. The new bill was tightened by Ahern and approved by Lighthizer before the council's final vote.